Let us consider the interactions of a flux of protons with energy \(E_p\)  with a target of hydrogen gas with density \(n_{\rm H}\)  (heavier target nuclei can approximately be accounted for by associating to them an effective H density, while for the generalization to the case of a flux of heavy nuclei see e.g. [1]}, [2]}). The neutrinos produced can be computed as
\(\frac{{\rm d}N_{\nu _i}}{{\rm d}E_\nu }=n_{\rm H}D\int _{E_\nu }^\infty \frac{{\rm d}E_p}{E_p}\sigma _{pp}F_{\nu _i}(x_\nu ,E_p)\frac{{\rm d}N_p}{{\rm d}E_p},\) 
